Power series

represents the coefficient of the nth term and c is a constant called the center of the series.

In many situations, the center c is equal to zero, for instance for Maclaurin series.

In such cases, the power series takes the simpler form

The partial sums of a power series are polynomials, the partial sums of the Taylor series of an analytic function are a sequence of converging polynomial approximations to the function at the center, and a converging power series can be seen as a kind of generalized polynomial with infinitely many terms.

Conversely, every polynomial is a power series with only finitely many non-zero terms.

Beyond their role in mathematical analysis, power series also occur in combinatorics as generating functions (a kind of formal power series) and in electronic engineering (under the name of the Z-transform).

The familiar decimal notation for real numbers can also be viewed as an example of a power series, with integer coefficients, but with the argument x fixed at 1⁄10.

One can view power series as being like "polynomials of infinite degree", although power series are not polynomials in the strict sense.

, is one of the most important examples of a power series, as are the exponential function formula

are not permitted; fractional powers arise in Puiseux series.

The set of the complex numbers such that |x – c| < r is called the disc of convergence of the series.

When two functions f and g are decomposed into power series around the same center c, the power series of the sum or difference of the functions can be obtained by termwise addition and subtraction.

The sum of two power series will have a radius of convergence of at least the smaller of the two radii of convergence of the two series,[2] but possibly larger than either of the two.

, the power series of the product and quotient of the functions can be obtained as follows:

Solving the corresponding equations yields the formulae based on determinants of certain matrices of the coefficients of

is given as a power series as above, it is differentiable on the interior of the domain of convergence.

A function f defined on some open subset U of R or C is called analytic if it is locally given by a convergent power series.

If a function is analytic, then it is infinitely differentiable, but in the real case the converse is not generally true.

This means that every analytic function is locally represented by its Taylor series.

The global form of an analytic function is completely determined by its local behavior in the following sense: if f and g are two analytic functions defined on the same connected open set U, and if there exists an element c ∈ U such that f(n)(c) = g(n)(c) for all n ≥ 0, then f(x) = g(x) for all x ∈ U.

The sum of a power series with a positive radius of convergence is an analytic function at every point in the interior of the disc of convergence.

However, different behavior can occur at points on the boundary of that disc.

For example: In abstract algebra, one attempts to capture the essence of power series without being restricted to the fields of real and complex numbers, and without the need to talk about convergence.

An extension of the theory is necessary for the purposes of multivariable calculus.

where j = (j1, …, jn) is a vector of natural numbers, the coefficients a(j1, …, jn) are usually real or complex numbers, and the center c = (c1, …, cn) and argument x = (x1, …, xn) are usually real or complex vectors.

is the set of ordered n-tuples of natural numbers.

More generally, one can show that when c=0, the interior of the region of absolute convergence is always a log-convex set in this sense.)

[4] Let α be a multi-index for a power series f(x1, x2, …, xn).

The order of the power series f is defined to be the least value

This definition readily extends to Laurent series.